I Matrix element involving raising/lowering operators

[Malamala]

Hello! I have this operator $Y_1^1(\theta,\phi)N_-$, where $N_-$ is the lowering operator for $|l,m>$. I want to calculate: $<0,0|Y_1^1(\theta,\phi)N_-|1,0>$ and I am getting:

$$<0,0|Y_1^1(\theta,\phi)N_-|1,0> = \sqrt{2}<0,0|Y_1^1(\theta,\phi)|1,-1> = \sqrt{2}\int Y_0^0(\theta,\phi)Y_1^1(\theta,\phi)Y_1^-1(\theta,\phi)\sin\theta d\theta d\phi = -\sqrt{2/3}$$

However, I can also apply the $N_-$ operator on the left, too, as a $N_+$ operator $(<0,0|N_+)Y_1^1(\theta,\phi)|1,0>$, which would give me zero, as $<0,0|N_+ = 0$. What am I doing wrong? I should get the same result whether I apply to the left or to the right, no?

I initially thought that there might be some commutation issues between $Y_1^1(\theta,\phi)$ and $N_{\pm}$, but here $Y_1^1(\theta,\phi)$ is number for each $(\theta,\phi)$, not a ket on which $N_{\pm}$ can act i.e. $Y_1^1(\theta,\phi) \neq |1,1>$, but $Y_1^1(\theta,\phi) = <\theta,\phi|1,1>$. So there should be no problem on moving $N_-$ around. I am really confused. What am I doing wrong?

 

[gentzen]

There should indeed be commutation issues, because $N_-$ is not diagonal in the position basis, but $Y_1^1(\theta,\phi)$ is, i.e. it is a simple multiplication in the position basis.

 

[Malamala]

There should indeed be commutation issues, because $N_-$ is not diagonal in the position basis, but $Y_1^1(\theta,\phi)$ is, i.e. it is a simple multiplication in the position basis.

But what should I do then? I assume that there should be a definite value for that matrix element, but I am not sure how to compute it.

 

[gentzen]

But what should I do then? I assume that there should be a definite value for that matrix element, but I am not sure how to compute it.

The computation of the non-zero result looks reasonable. No "assumed" commutativity was used there.

 

[Malamala]

The computation of the non-zero result looks reasonable. No "assumed" commutativity was used there.

Yes, but if I assume commutativity, I still don't get the right answer (it's non-zero, but still not right).

 

[gentzen]

Yes, but if I assume commutativity, I still don't get the right answer (it's non-zero, but still not right).

You mean "don't assume commutativity". And guess you have some "independent information", which allows you to know that your answer is not yet right.

But your question was just why your two different ways to compute the result gave different results. That question should have been clarified now.

 

[Malamala]

You mean "don't assume commutativity". And guess you have some "independent information", which allows you to know that your answer is not yet right.

But your question was just why your two different ways to compute the result gave different results. That question should have been clarified now.

No, sorry for the confusion. I don't have independent information. I know it's not correct as the 2 answers should agree (I don't know to what value, but that value should be the same for both).

So the fact that they still don't agree, even after using the non-commutativity, means that I still don't have an answer as to why they don't agree. I don't see how that clarifies my question...

Also, what do you mean by "You mean "don't assume commutativity"."?

 

[gentzen]

No, sorry for the confusion. I don't have independent information. I know it's not correct as the 2 answers should agree (I don't know to what value, but that value should be the same for both).

Let me try to clarify: Your first computation (which didn't rely on commutivity) gave $-\sqrt{2/3}$. Your second computation (which is wrong because it assumed commutivity) gave gave $0$. What I tried to clarify is why you should not be surprised that those two results are different.

Also, what do you mean by "You mean "don't assume commutativity"."?

I mean that I tried to make sense of what you wrote, and added the missing "don't". But apparently, the reason why your words don't make sense to me is different:

So the fact that they still don't agree, even after using the non-commutativity

You have not used the non-commutativity, or at least you have not shown how you did it, or reported your results.

 

[Malamala]

Let me try to clarify: Your first computation (which didn't rely on commutivity) gave $-\sqrt{2/3}$. Your second computation (which is wrong because it assumed commutivity) gave gave $0$. What I tried to clarify is why you should not be surprised that those two results are different.


I mean that I tried to make sense of what you wrote, and added the missing "don't". But apparently, the reason why your words don't make sense to me is different:

You have not used the non-commutativity, or at least you have not shown how you did it, or reported your results.

What I meant was that after you suggested that commutativity might be the issue I tried that and I still didn't get $-\sqrt{2/3}$, which I should. What I did was to apply the operator to the left which becomes:

$$(<0,0|Y_1^{-1})N_+ = (<0,0|N_+)Y_1^{-1} + <0,0|(Y_1^{-1}N_+) = \sqrt{2}<0,0|Y_1^0$$
So the matrix element becomes:

$$\sqrt{2}<0,0|Y_1^0|1,0> = \sqrt{2/3}$$
So I am still off by a minus sign. But honestly, I am not even sure if this is the right way to account for the fact that the 2 operators don't commute.

 

[Haborix]

This calculation is fraught from the start. As written, the matrix element is zero. If the answer is supposed to be non-zero, then the matrix element as written is wrong. By a sequence of questionable mathematical steps you get an answer that is not zero. The issues would probably resolve themselves if we started with the derivation of this matrix element.

 

[div_grad]

@Malamala you asked exactly the same question last week in the following thread (still up btw.):

https://www.physicsforums.com/threads/confused-about-a-matrix-element.1079973/

so I don't understand the point of starting a new thread instead of continuing the previous one? I've told you there that your operator, which you claimed to be Hermitian, is actually not Hermitian and this is one of the reasons for the "difference" between the two matrix elements you are calculating. I've also pointed out that the $N_\pm$ operators do not commute with the spherical harmonics.

Perhaps more importantly, in the previous thread you gave additional context (which is missing from this version of your question) that you are interested in these matrix elements for the case of a diatomic molecule. And for a diatomic molecule, the "raising/lowering" operators $N_\pm$ do not act in the same simple way as one is accustomed to from, e.g., the hydrogen atom case (i.e., a central potential). I've told you about this point in your previous thread as well.

It looks like you're trying to calculate matrix element of some kind of a "spin-orbit"- or "spin-rotation"-type interaction term, that is given by the scalar product of some two operators of the form $\mathbf{A}\cdot\mathbf{B}$. Can you give additional context as to what do you want to calculate here, and if this matrix element is supposed to pertain to the diatomic molecule case (as specified in your original thread)? Then we'll be able to help you clarify this issue.